# Generating collective counterfactual explanations in score-based classification via mathematical optimization

arXiv: https://arxiv.org/abs/2310.12822 · HTML: https://mlsanity.com/paper/2310.12822/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 54 references that are in the map.
- First posted on arXiv 19 Oct 2023.

## Builds on (most cited first, 27 of 27)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Explanation in Artificial Intelligence: Insights from the Social Sciences](https://mlsanity.com/paper/1706.07269/) (2017)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [Actionable Recourse in Linear Classification](https://mlsanity.com/paper/1809.06514/) (2018)
- [Explanations based on the Missing: Towards Contrastive Explanations with Pertinent Negatives](https://mlsanity.com/paper/1802.07623/) (2018)
- [FACE: Feasible and Actionable Counterfactual Explanations](https://mlsanity.com/paper/1909.09369/) (2019)
- [Model-Agnostic Counterfactual Explanations for Consequential Decisions](https://mlsanity.com/paper/1905.11190/) (2019)
- [Counterfactual Explanations and Algorithmic Recourses for Machine Learning: A Review](https://mlsanity.com/paper/2010.10596/) (2020)
- [Multi-Objective Counterfactual Explanations](https://mlsanity.com/paper/2004.11165/) (2020)
- [Towards Realistic Individual Recourse and Actionable Explanations in Black-Box Decision Making Systems](https://mlsanity.com/paper/1907.09615/) (2019)
- [Efficient Search for Diverse Coherent Explanations](https://mlsanity.com/paper/1901.04909/) (2019)
- [Best Subset Selection via a Modern Optimization Lens](https://mlsanity.com/paper/1507.03133/) (2015)
- [NICE: An Algorithm for Nearest Instance Counterfactual Explanations](https://mlsanity.com/paper/2104.07411/) (2021)
- [Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms](https://mlsanity.com/paper/1803.01454/) (2018)
- [The Intriguing Relation Between Counterfactual Explanations and Adversarial Examples](https://mlsanity.com/paper/2009.05487/) (2020)
- [Optimal Counterfactual Explanations in Tree Ensembles](https://mlsanity.com/paper/2106.06631/) (2021)
- [BRPO: Batch Residual Policy Optimization](https://mlsanity.com/paper/2002.05522/) (2020)
- [Semantics and explanation: why counterfactual explanations produce adversarial examples in deep neural networks](https://mlsanity.com/paper/2012.10076/) (2020)
- [Synthesizing Action Sequences for Modifying Model Decisions](https://mlsanity.com/paper/1910.00057/) (2019)
- [Scaling Guarantees for Nearest Counterfactual Explanations](https://mlsanity.com/paper/2010.04965/) (2020)
- [A Few Good Counterfactuals: Generating Interpretable, Plausible and Diverse Counterfactual Explanations](https://mlsanity.com/paper/2101.09056/) (2021)
- [Sparsity in Optimal Randomized Classification Trees](https://mlsanity.com/paper/2002.09191/) (2020)
- [DisCERN:Discovering Counterfactual Explanations using Relevance Features from Neighbourhoods](https://mlsanity.com/paper/2109.05800/) (2021)
- [Counterfactual Explanations Using Optimization With Constraint Learning](https://mlsanity.com/paper/2209.10997/) (2022)
- [Optimal Counterfactual Explanations for Scorecard modelling](https://mlsanity.com/paper/2104.08619/) (2021)

## Built on by (most cited first, 9 of 9)

- [GLANCE: Global Actions in a Nutshell for Counterfactual Explainability](https://mlsanity.com/paper/2405.18921/) (2024)
- [FACEGroup: Feasible and Actionable Counterfactual Explanations for Group Fairness](https://mlsanity.com/paper/2410.22591/) (2024)
- [One-for-many Counterfactual Explanations by Column Generation](https://mlsanity.com/paper/2402.09473/) (2024)
- [Collective Counterfactual Explanations: Balancing Individual Goals and Collective Dynamics](https://mlsanity.com/paper/2402.04579/) (2024)
- [Learning Actionable Counterfactual Explanations in Large State Spaces](https://mlsanity.com/paper/2404.17034/) (2024)
- [Understanding Fixed Predictions via Confined Regions](https://mlsanity.com/paper/2502.16380/) (2025)
- [Revealing Positive and Negative Role Models to Help People Make Good Decisions](https://mlsanity.com/paper/2603.02495/) (2026)
- [A Unified Optimization Framework for Multiclass Classification with Structured Hyperplane Arrangements](https://mlsanity.com/paper/2510.05047/) (2025)
- [Optimal Transport Group Counterfactual Explanations](https://mlsanity.com/paper/2601.20692/) (2026)

## Abstract (arXiv)

Due to the increasing use of Machine Learning models in high stakes decision making settings, it has become increasingly important to have tools to understand how models arrive at decisions. Assuming a trained Supervised Classification model, explanations can be obtained via counterfactual analysis: a counterfactual explanation of an instance indicates how this instance should be minimally modified so that the perturbed instance is classified in the desired class by the Machine Learning classification model. Most of the Counterfactual Analysis literature focuses on the single-instance single-counterfactual setting, in which the analysis is done for one single instance to provide one single explanation. Taking a stakeholder's perspective, in this paper we introduce the so-called collective counterfactual explanations. By means of novel Mathematical Optimization models, we provide a counterfactual explanation for each instance in a group of interest, so that the total cost of the perturbations is minimized under some linking constraints. Making the process of constructing counterfactuals collective instead of individual enables us to detect the features that are critical to the entire dataset to have the individuals classified in the desired class. Our methodology allows for some instances to be treated individually, performing the collective counterfactual analysis for a fraction of records of the group of interest. This way, outliers are identified and handled appropriately. Under some assumptions on the classifier and the space in which counterfactuals are sought, finding collective counterfactuals is reduced to solving a convex quadratic linearly constrained mixed integer optimization problem, which, for datasets of moderate size, can be solved to optimality using existing solvers. The performance of our approach is illustrated on real-world datasets, demonstrating its usefulness.